The previous discussion shows that under Assumptions REF , REF , and REF , after re-arranging the filter equation to (REF ), to solve the projection filter, it remains to calculate \(\psi \) . In this section, we will describe how to compute \(\psi \)  efficiently for the unidimensional case (\(d_x=1\) ), while the next section covers the multidimensional case. We will start by defining a bijection that will be used in the numerical integration of the log-partition function (REF ). In the most general setting, the support of the parametric probability \(p(\cdot , \theta )\)  depends on \(\theta \) . However, [1]} pointed out that this condition poses a substantial difficulty in analysis. Therefore, following [1]}, we assume that the support of \(p(\cdot ,\theta )\)  is uniform for all \(\theta \in \Theta \) . For notational simplicity let us set \(\Omega = (-1,1)\) , and denote the class of bounded variation functions on \(\Omega \)  as \(BV(\bar{\Omega })\) , where \(\bar{\Omega }\)  is the closure of \(\Omega \) .
